Problem: Let $f$ be a transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$. Its Jacobian matrix is given below. $J(f) = \begin{bmatrix} \dfrac{1}{x} & 0 \\ \\ 1 & 1 \end{bmatrix}$ Find the Jacobian determinant of $f$. $|J(f)| = $ How will $f$ expand or contract space around the point $\left( \dfrac{1}{9}, \dfrac{1}{3} \right)$ ? Choose 1 answer: Choose 1 answer: (Choice A) A Leave it the same (Choice B) B Expand it finitely (Choice C) C Contract it finitely (Choice D) D Contract it infinitely
The Jacobian determinant is the determinant of the Jacobian matrix. It represents the factor by which the transformation $f$ expands or contracts volume around a certain input. $\begin{aligned} |J(f)| &= \det \left( \begin{bmatrix} \dfrac{1}{x} & 0 \\ \\ 1 & 1 \end{bmatrix} \right) \\ \\ &= \dfrac{1}{x} \end{aligned}$ If we evaluate $|J(f)|$ at $\left( \dfrac{1}{9}, \dfrac{1}{3} \right)$, we get $9$. Because the Jacobian determinant here has an absolute value greater than $1$, we can conclude that $f$ will finitely expand the space around $\left( \dfrac{1}{9}, \dfrac{1}{3} \right)$. To recap, the Jacobian determinant of $f$ is $\dfrac{1}{x}$, and $f$ will finitely expand the space around the point $\left( \dfrac{1}{9}, \dfrac{1}{3} \right)$.